Archimedes placed hexagons within and around a circle, and he reasoned that pi has a value between the perimeters of the inner and outer hexagon. Not stopping there, Archimedes proceeded to 96-sided polygons, showing that 223/71 ≤ π ≤ 22/7; that is, 3.1408 ≤ π ≤ 3.1429. If we average these, we get 3.141855, which is within a hundredth of a percent of the value of pi. I use just the first eleven digits of pi, 3.1415926535, in most of my computer programs. That last 5 should be rounded up to 6, but I don't bother. A single-precision (float) in C is precise to about 7 digits.

Examination of any segment of the digits of pi suggests that its digits are random. A closer look using computer techniques reaffirms this idea. It's conjectured that pi is a normal number; that is, an irrational number whose digits occur with the same likelihood. This is true in every number base, not just our commonly used base 10. If pi were normal, then none of its digit, or any combination of digits, occurs more frequently than any other. Digits of pi are used as random numbers in the Blowfish cipher.

While it's possible to do improved statistical tests of the distribution of the digits of pi, since more and more of its digits are being amassed, there's no real proof of pi's normality. Not only have people looked at the distribution of the digits, themselves, but they've looked at the distribution of pairs of digits (dyads), groups of three digits (triads), up to larger n-ads. All these tests indicate a normal number. While it's almost certainly true that pi and its irrational cousins, √2 and e (the base of natural logarithms) are normal numbers, no mathematician has actually proved this.

The fractal analysis would detect whether there are complex structures that exist in the sequence of pi's digits, since it looks at the sequence as a whole, and not just its parts. In this analysis, the digits of pi are considered to be periodic samples of a waveform. For this analysis, Sevcik generated his own list of digits of pi using a variant of the Ramanujan series developed by the famed Chudnovsky brothers. To illustrate how far computing has progressed, this operation took just 1929 seconds on his Linux computer.[2]

While Sevcik's result is just another statistical argument for the normality of pi, his results, as shown in the graph, are convincing. His fractal analysis demonstrates that the digits of pi and a sequence of random numbers show the same fractal behavior.

Fractal dimension of the digits of pi (circles) compared with that of a sequence of random integers (triangles) for sequences up to a billion (109).